A classic problem in quantum mechanics is a one-dimensional well with infinite walls. Such a system can be used to describe the behavior of electrons in a very thin metal film.

Consider the following potential:
\[U(x) = \begin{cases}
+\infty, \quad &x<0 \text{ or } x > a \\
0, \quad & 0 < x < a
\end{cases},\]
where $a$ is the width of the well.

For such a system, the probability of finding a particle outside the boundaries of the well is zero, and therefore $\Psi(0) = \Psi(a) = 0$.

A1 Find the solutions $\Psi_n$ of the Schrödinger equation:
\[ -\frac{\hbar^2}{2m} \Psi'' + U(x) \Psi = E \Psi.\]Write down corresponding energies $E_n$.

Let's imagine that there are $N$ electrons in the well that dont interact with each other (for example the elecrtostatic interaction is screened in the metal). Remember that electrons are fermions and that's why they cannot be in the same state.

A2 Calculate the energy $E$ of the whole electrons.

Fermions cannot be in the same state and that's why part of elecrtons <> to be on the levels with higher energy. From the classial point of view it's interaction, which calls exchange interaction.

A3 Determine the force $F$ of pressure of electons on the each wall of the well.